Nonlinear modeling of area and production of sugarcane in Tamil Nadu, India

The present investigation was carried out to model the trend of area and production of sugarcane in Tamil Nadu. It was obtained by using the secondary data of area and production over a period of 30 years (1984-85 to 2014-15). For this purpose, Different nonlinear models such as Logistic, Gompertz, Rational, Gaussian, Weibull, Hoerl and Sinusoidal models were employed. Levenberg-Marquardt technique was used to obtain the estimates of the unknown parameters of the nonlinear regression models.

Original Research Article https://doi.org/10.20546/ijcmas.2018.710.363

Nonlinear Modeling of Area and Production of

Sugarcane in Tamil Nadu, India

P Dinesh Kumar * , Bishvajit Bakshi and V Manjunath

Department of Agricultural Statistics, Applied Mathematics and Computer Sciences, UAS,

GKVK, Bengaluru-65, Karnataka, India

*Corresponding author

Introduction

Sugarcane, a traditional crop of India plays an

important role in agricultural and industrial

economy of the country It is cultivated in

most of the states and though it covers an

insignificant share in gross cropped area of the

country, its share in the country’s economic

growth has become significant The crop is

grown in more than 120 countries, of which,

Brazil (736 million tonnes), India (352 million

tonnes) and China (126 million tonnes) are the

top three countries in production (Anon,

2015) In 2015, Uttar Pradesh recorded the

highest area of sugarcane of about 42.25 per cent, followed by Maharashtra (20.33%), Karnataka (9.47%), Tamil Nadu (5.19%), Gujarat (4.11%) and Andhra Pradesh (2.74%) contributing about 84 per cent of the total area

in India Currently in Tamil Nadu, 0.263 million hectares are under cane cultivation and this is increasing annually due to the increased consumption of sugar and also the growing demand from mills for sugar cane as a raw material Because of its diversified uses in different industries, this crop is considered as

‘‘Karpagavirucham’’ and in modern

terminology as ‘‘wonder cane’’ (Mohan et al.,

International Journal of Current Microbiology and Applied Sciences

ISSN: 2319-7706 Volume 7 Number 10 (2018)

Journal homepage: http://www.ijcmas.com

The present investigation was carried out to model the trend of area and production of sugarcane in Tamil Nadu It was obtained by using the secondary data of area and production over a period of 30 years (1984-85 to 2014-15) For this purpose, Different nonlinear models such as Logistic, Gompertz, Rational, Gaussian, Weibull, Hoerl and Sinusoidal models were employed Levenberg-Marquardt technique was used to obtain the estimates of the unknown parameters of the nonlinear regression models To select a best fitted model for the area and production of sugarcane in Tamil Nadu, the model adequacy statistics such R2, RMSE, MAE and residual assumption tests such as Runs test, Shapiro-Wilks test and Durbin-Watson test were carried out For area of sugarcane, it was found that Logistic model had the lowest Root Mean Square Error (27.770), Mean Absolute Error (18.737) and the highest R2 value (74.7 per cent) Hence, Logistic model is the most suitable among the fitted nonlinear model which can be used for further trend analysis on the area under sugarcane For production of sugarcane, Gaussian model had the lowest Root Mean Square Error (2.604), Mean Absolute Error (2.760) and the highest R2 value (78.2 per cent) Hence, Gaussian model is the most suitable among the fitted nonlinear model which can be used for further trend analysis on the production of sugarcane

K e y w o r d s

Nonlinear models, R2,

Root mean square error,

Mean absolute error,

Durbin-Watson statistic,

Levenberg-Marquardt

technique, Shapiro-Wilks

statistic

Accepted:

24 September 2018

Available Online:

10 October 2018

Article Info

2007) From the above justified facts, it is

evident that there is a considerable scope to

study the trend in area and production of

sugarcane crop in Tamil Nadu

Materials and Methods

The present study is conducted with the

overall objective of estimating suitable

regression model that explains the trend of

area and production of sugarcane in Tamil

Nadu For this study, A secondary data of

area, production and productivity of sugarcane

in Tamil Nadu for the period of 30 years from

1985 to 2014 were collected from the

Department of Economics and Statistics,

Government of Tamil Nadu

Non-linear regression models

Statistical modelling essentially consists in

constructing a model, represented by a set of

equations to describe the input-output

relationship among the variables of interest

From a realistic point of view, such a

relationship among variables in agriculture

and biological sciences is ‘nonlinear’ in

nature In such a model, a unit increase in the

value of independent variable(s) may not

result in an equivalent unit increase in the

dependent variable A nonlinear regression

model is one in which at least one of the

parameters appears nonlinearly A nonlinear

model, which can be transformed into linear

model by some transformation is called

‘intrinsically linear’, else it is called as

‘intrinsically nonlinear’ Mathematically, in

nonlinear models at least one of the

derivatives of the expectation function with

respect to at least one parameter is a function

of parameter(s) The model is a nonlinear

regression model as the derivatives of Y with

respect to a and b are both functions of a and /

or b Like in linear regression, parameters in a

nonlinear model can also be estimated by the

method of least squares However, due to the

difficulty in the procedure of computation, the common practice is to work with the log transformed model

x

Y  a b  e

The log transformation is valid only when

error term ‘e’ in the above equation is

multiplicative in nature Thereafter, method of least square is used to estimate the unknown parameters Furthermore, R2 value is calculated to measure the goodness of fit of the model

The log transformed procedure suffers from some important drawbacks

Original structure of the error term got disturbed due to transformation

R2 values computed, assess the goodness of fit

of the transformed model and not of the original nonlinear model

Proceeding further to carryout residual analysis for the residuals generated by the transformed model, will result in erroneous conclusion

As a remedy to these pitfalls, nonlinear regression procedures are already developed

in literature which necessitates computer intensive tools to find solution for the parameters (Venugopalan and Shamasundaran, 2003) The following nonlinear models are considered in the present investigation

Where Y is the area/production during the time

X; A, B, C and D are the parameters, and ‘e’ is

the error term The parameter ‘C’ is the intrinsic growth rate and the parameter ‘A’

represents the carrying capacity for each

model Symbol ‘B’ represents different functions of the initial value Y(0) and ‘B’ is

the added parameter In addition to the above

nonlinear models some other nonlinear models

also are employed as per the data need

To obtain estimates of the unknown

parameters of a nonlinear regression model,

Levenberg-Marquardt technique was used In

this method, the following steps are carried

out

Step I: Starting with a good initial guess of the

unknown parameters, a sequence of θ’s which

hopefully converge to θ is computed

Step II: Error sum of squares or objective

function expressed as

2

1

n

x



 

is minimized with respect to the current value

of θ The new estimates are obtained

Step III: By feeding the recently obtained

estimates as the initial guess for the next

iteration, objective function S(θ) is minimized

again to obtain fresh estimates This procedure

is continued till the successive iteration

yielded parameter estimate values are close to

each other

Choice of starting values of the parameters

for various models

All the iterative procedures require initial

values θ r0 (r = 1, 2, 3…, k) of the parameter θ r

The choice of good initial values can spell the

difference between success and failure in

locating the fitted value or between rapid and

slow convergence to the solution Also, if

multiple minima exist in addition to absolute

minimum, poor starting values may result in

convergence to an unwanted stationary point

of the sum of squares surface This unwanted

point may have parameter values which are

physically impossible or which does not

provide the true minimum value of S(θ).There

are number of ways to determine initial

parameter values for nonlinear models The

most obvious method for making the initial guesses is by the use of prior information Estimates calculated from previous experiments, known values from similar systems, values computed from theoretical considerations: all these form ideal initial guesses In this study the Curve expert Ver.1.3 software package is used to estimate the initial values

Model adequacy checking

To test the goodness of fit of the fitted polynomial model, the co-efficient of

determination R 2 defined as the proportion of total variation in the response variable (time) being explained by the fitted model is widely used

2

2

ˆ

1

1

n

i

n

i i













To test the overall significance of the model, the F test is used

2

2 1 1

R k F

R

n k

 

 

 



  

   

Which follows F distribution with k (number

of parameter in the model), (n-k-1) degrees of

freedom

Adjusted R2 is a modification of R2 that adjusts for the number of explanatory terms in

a model Unlike R2, the adjusted R2 increases only if the new term improves the model more than would be expected by chance The

adjusted R2 can be negative and will always be

less than or equal to R2 The adjusted R2 is defined as

2 2 ( 1)

n

n k



  

 

Where,

‘k’ is the number of parameters in the equation

‘n’ is the is the total number of observations

In addition to the above, two more reliability

statistics viz., Root Mean Square Error

(RMSE) and Mean Absolute Error (MAE) are

generally utilized to measure the adequacy of

the fitted model and it can be computed as

follows:

2

1 2 ˆ

1

n

Y Y

i

RMSE

n





ˆ -1

n

Y Y

i MAE

n







The lower the values of these statistics, the

better are the fitted model

Assumptions of error term

An important assumption of nonlinear

regression is that the residual ‘ε’, or the

dependent variable ‘Y’ follows normal

distribution

This assumption is required for test of

hypothesis about the regression coefficients

This assumption was verified using,

Shapiro-Wilk test was used to test for

normality The test statistic value of ‘W’

ranges from 0 to 1 When W = 1 the given data

are perfectly normal in distribution (Shapiro,

et al., 1968)

When ‘W’ is significantly lesser than 1, the

assumption of normality is not met The test

statistic is

 

2

1

2

1 ( - )

n

i n

i

a x W





 



Where,x i

is the ith order statistic, i.e., the ith

smallest number in the sample;x is the sample mean; The constants a i are given by

1 -1

T n

T

m V

a a a

m V V m



Where,

 1, 2, , T

T

n

andm m1, 2, ,m nare

the expected values of the order statistics of independent and identically-distributed random variables sampled from the standard

normal distribution, and V is the covariance matrix of those order statistics Then values a i, coefficients are tabulated by Shapiro and Wilk (1965)

Durbin-Watson test is used to test the presence

or absence of autocorrelation in residuals Durbin-Watson is the ratio of the distance between the errors to their overall variance The test statistic is

2 -1 2

2

1

n

i n i i

d

e





2 (1-  )



Where ei  yi y ˆi

and y i

and yˆi

are, respectively, the observed and predicted values of the response variable for individual

i Thus, DW is equal to 2 minus two times the

correlation of e t and e t-1 Durbin-Watson is used both as diagnostic for

autocorrelation and as estimate of ρ DW

statistic is a correlation and thus depends on

values of independent variables as -1 ≤ ρ ≤ +1

thus 0 ≤ DW ≤ 4

The runs test (Bradley, 1968) was used to

decide if a data set is from a random process

The test statistics is

~ (0,1)

r

r

r







Where, Mean

1 2

1 2

2

1

r

n n

n n

1 2 1 2 1 2

2

( )

r

With n 1 and n 2 denoting the number of

positive and negative values in the series

respectively

The runs test rejects the null hypothesis, if

1

2





Results and Discussion

Three parameter mechanistic growth models

such as Logistic, Gompertz, Gaussian and

Hoerl models, and four parameter mechanistic

growth models such as Ration function,

Weibull and Sinusoidal models were used for

studying area and production of sugarcane in

Tamil Nadu The Levenberg-Marquardts

procedure is the most efficient iteration

procedure described in the methodology,

which was used for solving nonlinear normal

equations The results are discussed in the

followings

Model based trend analysis for area under

sugarcane in Tamil Nadu

For the area under the cultivation of

sugarcane, the nonlinear models such as

Logistic, Rational, Gompertz, Sinusoidal and

Weibull models were fitted which were

graphically represented in the Figure 1 and 2

The results presented in the Table 1 which

reveals that, among the different nonlinear

models fitted, the maximum R2 value of 74.7

per cent was observed in the logistic model with the minimum values of RMSE (27.770) and MAE (18.737) on comparison with all other nonlinear models The next best nonlinear model was the Rational model with

73 per cent of R2 value

The p value of Shapiro-Wilks test statistic

(0.920) and the Run test statistic (0.436) indicates that the residuals of the logistic model were normal and random respectively The Durbin-Watson statistic recorded the value of 1.577, which indicated that there was

no serial correlation among the residuals and were independent The scatter diagram and normal plot for the residuals of the logistic model confirmed those assumptions

For the best fitted logistic model, all the model coefficients were highly significant at 1 per cent The parameter estimates of the logistic model were with a carrying capacity of 322.627 and the intrinsic growth rate of 0.176

Among the nonlinear models fitted for the area under sugarcane, obtained suitable logistic function was as follows,

322.627 ˆ

1 1.003 exp 0.176

Y

X



R2 = 74.7 per cent

Model based trend analysis for production

of sugarcane in Tamil Nadu

For the production of sugarcane, the nonlinear models such as Logistic, Rational, Gompertz, Sinusoidal, Weibull and Gaussian models were fitted which were graphically represented in the Figure 3 and 4 The results presented in the Table 2 revealed that, among the different nonlinear models fitted, the maximum R2 value of 78.2 per cent was observed in the Gaussian model with the minimum RMSE (2.604) and MAE (2.760) values on comparison with all other nonlinear models

Fig 3.1: Graph of the actual values and fitted models for the area

undersugarcane in Tamil Nadu

Fig 3.2: Graph of the actual values and fitted models for the area under

sugarcane in Tamil Nadu

Fig 3.3: Graph of the actual values and fitted models for the production of

sugarcane in Tamil Nadu

Fig 3.4: Graph of the actual values and fitted models for the production

of sugarcane in Tamil Nadu

Table.2 Estimates of the parameters along with model adequacy of fitted nonlinear models for area under sugarcane (1985-2014)

Logistic Gompertz Rational Sinusoidal Weibull Gaussian

Carrying Capacity / Intercept (A) 322.627** 324.796** 174.628** 279.756** 315.601** 331.641**

(11.638) (13.230) (20.208) (9.107) (8.659) (8.279)

Function of initial value (B) 1.003** -0.324 4.617 44.373** 124.032** 21.916**

(0.226) (0.177) (9.189) (12.866) (22.722) (1.455)

Intrinsic growth rate / slope (C) 0.176** 0.147** -0.028 1.091** 0.006 0 190

(0.049) (0.045) (0.027) (0.033) (0.015) (2.135)

(0.0001) (0.594) (1.093)

* Significant at 5% level; ** Significant at 1% level

RMSE: Root Mean Square Error; MAE: Mean Absolute Error

Values in parentheses () indicate standard errors

Table.3 Estimates of the parameters along with model adequacy of fitted nonlinear models for production of sugarcane (1985-2014)

Logistic Gompertz Rational Sinusoidal Hoerl Gaussian

Carrying Capacity / Intercept (A) 32.885** 33.090** 17.009* 30.365** 18.122** 34.735**

(1.803) (2.018) (8.181) (0.889) (2.608) (0.814)

Function of initial value (B) 0.845* -0.456 4.088 -5.482** 0.992** 21.620**

(0.410) (0.385) (10.044) (1.292) (0.008) (1.296)

Intrinsic growth rate / slope (C) 0.202 0.169 0.093 1.054** 0.247* 0.236**

(0.117) (0.104) (0.330) (0.027) (0.099) (2.027)

(0.002) (0.491)

* Significant at 5% level; ** Significant at 1% level

RMSE: Root Mean Square Error; MAE: Mean Absolute Error

Values in parentheses () indicate standard errors

Table.1 Nonlinear regression models

S No Name of the model Model

I Logistic model

A

B C X

II Gompertz Relation model Y  Aexp expB C X  e

III Rational Function

2

1

A B X



2

2

C

 

-D

C X

VII Sinusoidal model Y A Bcos (C X D- ) e

The next best nonlinear model was the Hoerl

model with 49.7 per cent of R2 value

The p value of Shapiro-Wilk test statistic

(0.186) and the Run test statistic (0.993) to

test for assumptions indicates that the

residuals of the Gaussian model were normal

and random respectively The Durbin-Watson

statistic recorded the value of 2.247 which

indicated that there was no serial correlation

among the residuals and was independent

The scatter diagram and normal plot for the

residuals of the Gaussian model in support of

numerical test confirmed the liability of the

residual assumptions (Table 1–4)

For the best fitted Gaussian model, all the

coefficients were showing significant at 1 per

cent level of significance The parameter

estimates of the Gaussian model were with a

carrying capacity of 34.735 and the intrinsic

growth rate of 0.236

Gaussian model which was found to be the

suitable model for the production of

sugarcane is as follows,

34.735 exp (21.620 )

ˆ

0.111

X



R2 = 78.2 per cent

Sugarcane is one of the important cash crops

in Tamil Nadu Due to the climatic and many other reasons, there is a lot of fluctuations in the area and production of Tamil Nadu So, there is a necessity to study the trend in area and production of sugarcane and the impact of precipitation on the productivity of sugarcane

in different agro-climatic zones It was observed that nonlinear models are more appropriate to visualize the temporal trend of area and production of sugarcane in Tamil Nadu Logistic and Gaussian models were the most suitable fitted models which clearly explained the trend of area and production of sugarcane in Tamil Nadu

References

Anonymous, 2015.Food and Agriculture

statistics 2015 Food and Agriculture

Organisation of United Nations, Rome, Italy http://www.fao.org/fao stat/en/#data

Bradley, J V., 1968 Distribution-free

Statistical Tests Prentice-Hall, Englewood Cliffs, NJ, USA

Mohan, S., Rajendran, K., Sivam, D and Saliha, B., 2007 Sugar –The wonder